CP Electrons go through a single slit 300 nm wide and strike a screen 24.0 cm away. At angles o

step 1 Okay, we have another electron diffraction problem. And in this question, we're told that electr

CP Electrons go through a single slit 300 nm wide and strike...

CP Electrons go through a single slit $300 \mathrm{nm}$ wide and strike a screen $24.0 \mathrm{~cm}$ away. At angles of $\pm 20.0^{\circ}$ from the center of the diffraction pattern, no electrons hit the screen, but electrons hit at all points closer to the center. (a) How fast were these electrons moving when they went through the slit? (b) What will be the next pair of larger angles at which no electrons hit the screen?

Key Concepts

Single-Slit Diffraction

Single-slit diffraction explains how waves spread out after passing through a narrow aperture. The interference patterns formed, including the positions of dark and bright fringes, are governed by the relationship a sin(?) = m?, where a is the slit width, ? is the diffraction angle, ? is the wavelength of the wave, and m is a nonzero integer. This principle is also applied to matter waves such as electrons when analyzing their diffraction patterns.

Relationship Between Momentum and Velocity

In the context of quantum mechanics, the momentum of a particle is related to its mass and velocity by p = mv. By combining this with the de Broglie wavelength formula, one can determine a particle's velocity from its observed diffraction behavior. This relationship is key in experiments where the wave properties of matter are used to infer classical quantities like speed.

de Broglie Wavelength

This concept concerns the wave-like behavior of particles, stating that any moving particle has an associated wavelength given by ? = h/p, where h is Planck's constant and p is the momentum of the particle. This idea is fundamental in quantum mechanics as it bridges the gap between particle and wave descriptions of matter.

Transcript

00:01 Okay, we have another electron diffraction problem.

00:03 And in this question, we're told that electrons move through a slit, which is with a three times a minute, and the first angle where we see no electrons throwing up on a screen on the slit is 20 degrees, which is both above and below the horizontal.

00:25 And the distance from the split to the screen is 24 centimeters.

00:29 So in question a, they ask us, what is the velocity of the electrons? so we want to start by finding a way to relate the information that we have to velocity.

00:42 So right now we have theta and d and l.

00:48 One equation we can use is d times sine theta is equal to lambda.

00:56 And then we can use lambda to find an expression for v.

01:01 Since we know that the divorily wavelength, lambda equals h over mv, depends on velocity.

01:09 So we can just plug this in up here, since these are equal.

01:14 So d -sign theta is equal to h over mv.

01:20 So now we just have to solve for the velocity.

01:23 So we're going to multiply it to the other side and then divide all this stuff down.

01:28 So we're going to get velocity is equal to h over d, sine theta times m.

01:37 All right, so all these things are known.

01:41 Just plug into velocity for an electron.

01:43 Theta is going to be 20 degrees.

01:45 D is the width of the split.

01:47 And of course, h is a constant.

01:49 So we get v is equal to 6 .946 times 10 to the 3 meters per second.

02:00 Okay.

02:00 Okay, and then in question b, they ask us, what is the second angle, which i'll call theta 2, where we don't see any electrons showing up on the screen? so the first gap was right here between this main peak and the smaller peak.

02:20 The next gap is going to be where the smaller peak ends before a third even smaller peak would start, or alternatively down here.

02:29 So we're looking for the 2, which is this angle.

02:37 Okay, so one thing we know about diffraction patterns is that the distance between the peaks is constant.

02:45 So this distance is going to be the same as this distance.

02:52 So if we can find out what this distance is, i'll call it y1.

02:55 Then we can determine the total distance, i'll call it y2 from the distance...

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